scholarly journals On a Conjecture of Livingston

2017 ◽  
Vol 60 (1) ◽  
pp. 184-195 ◽  
Author(s):  
Siddhi Pathak
Keyword(s):  
Linear Independence ◽  
Arithmetical Function ◽  
Algebraic Numbers ◽  
Erdös Conjecture

AbstractIn an attempt to resolve a folklore conjecture of Erdös regarding the non-vanishing at s = 1 of the L-series attached to a periodic arithmetical function with period q and values in {−1, 1}, Livingston conjectured the -linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston’s conjecture for composite q ≥ 4, highlighting that a newapproach is required to settle Erdös conjecture. We also prove that the conjecture is true for prime q ≥ 3, and indicate that more ingredients will be needed to settle Erdös conjecture for prime q.

scholarly journals Distribution and non-vanishing of special values of L-series attached to Erdős functions

2019 ◽  
Vol 15 (07) ◽  
pp. 1449-1462
Author(s):  
Siddhi S. Pathak
Keyword(s):  
Positive Integer ◽  
Characteristic Function ◽  
Arithmetical Function ◽  
Limiting Distribution ◽  
Special Values ◽  
Erdös Conjecture

In a written correspondence with A. Livingston, Erdős conjectured that for any arithmetical function [Formula: see text], periodic with period [Formula: see text], taking values in [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text], the series [Formula: see text] does not vanish. This conjecture is still open in the case [Formula: see text] or when [Formula: see text]. In this paper, we obtain the characteristic function of the limiting distribution of [Formula: see text] for any positive integer [Formula: see text] and Erdős function [Formula: see text] with the same parity as [Formula: see text]. Moreover, we show that the Erdős conjecture is true with “probability” one.

scholarly journals ON THE ℤp-RANKS OF TAMELY RAMIFIED IWASAWA MODULES

2013 ◽  
Vol 09 (06) ◽  
pp. 1491-1503 ◽  
Author(s):  
TSUYOSHI ITOH ◽  
YASUSHI MIZUSAWA ◽  
MANABU OZAKI
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Rational Number ◽  
Quadratic Field ◽  
Linear Independence ◽  
Prime Numbers ◽  
Imaginary Quadratic Field ◽  
Algebraic Numbers ◽  
Finite Set ◽  
Explicit Formulae

For a finite set S of prime numbers, we consider the S-ramified Iwasawa module which is the Galois group of the maximal abelian pro-p-extension unramified outside S over the cyclotomic ℤp-extension of a number field k. In the case where S does not contain p and k is the rational number field or an imaginary quadratic field, we give the explicit formulae of the ℤp-ranks of the S-ramified Iwasawa modules by using Brumer's p-adic version of Baker's theorem on the linear independence of logarithms of algebraic numbers.

ELLIPTIC CURVES AND -ADIC ELLIPTIC TRANSCENDENCE

2021 ◽  
pp. 1-6
Author(s):  
DUC HIEP PHAM
Keyword(s):  
Elliptic Curves ◽  
Short Proof ◽  
Linear Independence ◽  
Elliptic Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Algebraic Numbers ◽  
Algebraic Dependence ◽  
Necessary And Sufficient

Abstract We prove a necessary and sufficient condition for isogenous elliptic curves based on the algebraic dependence of p-adic elliptic functions. As a consequence, we give a short proof of the p-adic analogue of Schneider’s theorem on the linear independence of p-adic elliptic logarithms of algebraic points on two nonisogenous elliptic curves defined over the field of algebraic numbers.

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